I my book, it says: To prove that a set is denumerable you should find a bijection .

Now, I need to prove that is denumerable.
So I replace my with in the previous paragraph. Now I need to find a bijection .

Would it be equivalent to find a bijection ?
If so, I need to find a function that takes the rational numbers to . Also, it need to be injective.
It should be suffiecient to find a bijection ?

What could look like?

no it would not be equivalent to find a bijection becuase Q contains N so a trivial function would be . You need to find a function . Also bijection means both injection and surjection. so u don't need to single out injective.

for example, to prove that is denumerable. notice you can break into classes depending on what the numerator and denominator adds up to like class{2} = 1/1, class{3} = 1/2, 2/1 class{4} = 3/1, 1/3 etc.... and each of these classes is finite...

Consider the cross product of positive integers, .
Map that set to by
It can be shown that is a bijection.
The positive rational can be seen as a subset of .
So they are denumerable. Clearly we can extend this to the entire set of rationals.